Cyclic position codes

ABSTRACT

A method of determining an offset, modulo n, of at least one point with respect to a sequence  50  of at least n symbols, the sequence consisting of a repeating codeword of a cyclic position code, the cyclic position code having length n and minimum distance d min , the method including: obtaining, from the sequence  50  and at a position corresponding to the at least one point, a subsequence  52  of length w symbols, where w≧n−d min   +1;  mapping the subsequence  52  to a codeword  58  of the cyclic position code most likely to match the subsequence in the presence of symbol errors  60, 62  in the subsequence; and determining an offset  54 , in the sequence  50 , of the codeword  58  thus obtained, and thereby determining the offset of the at least one point.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is a 371 of PCT/AU01/01274 filed on Oct. 11,2001.

FIELD OF INVENTION

This invention relates to error-detecting and error-correcting cyclicposition codes and their use in the position-coding of surfaces.

CO-PENDING APPLICATIONS

Various methods, systems and apparatus relating to the present inventionare disclosed in the following co-pending applications filed by theapplicant or assignee of the present invention on 27 Nov. 2000:

PCT/AU00/01442, PCT/AU00/01444, PCr/AU00/01446, PCT/AU00/01445,PCT/AU00/01450, PCT/AU00/01453, PCr/AU00/01448, PCT/AU00/01447,PCT/AU00/01459, PCT/AU00/01451, PCr/AU00/01454, PCT/AU00/01452,PCT/AU00/01443, PCT/AU00/01455, PCT/AU00/01456, PCT/AU00/01457,PCT/AU00/01458 and PCT/AU00/01449.

The disclosures of these co-pending applications are incorporated hereinby cross-reference.

Various methods, systems and apparatus relating to the present inventionare disclosed in the following co-pending applications filed by theapplicant or assignee of the present invention on 20 Oct. 2000:

PCT/AU00/01273, PCT/AU00/01279, PCT/AU00/01288, PCT/AU00/01282,PCT/AU00/01276, PCT/AU00/01280, PCT/AU00/01274, PCT/AU00/01289,PCT/AU00/01275, PCT/AU00/01277, PCT/AU00/01286, PCT/AU00/01281,PCT/AU00/01278, PCT/AU00/01287, PCT/AU00/01285, PCT/AU00/01284 andPCT/AU00/01283.

The disclosures of these co-pending applications are incorporated hereinby cross-reference.

Various methods, systems and apparatus relating to the present inventionare disclosed in the following co-pending applications filed by theapplicant or assignee of the present invention on 15 Sep. 2000:PCT/AU00/01108, PCT/AU00/01110 and PCT/AU00/01111. The disclosures ofthese co-pending applications are incorporated herein bycross-reference.

Various methods, systems and apparatus relating to the present inventionare disclosed in the following co-pending applications filed by theapplicant or assignee of the present invention on 30 Jun. 2000:PCT/AU00/00762, PCT/AU00/00763, PCT/AU00/00761, PCT/AU00/00760,PCT/AU00/00759, PCT/AU00/00758, PCT/AU00/00764, PCT/AU00/00765,PCT/AU00/00766, PCT/AU00/00767, PCT/AU00/00768, PCT/AU00/00773,PCT/AU00/00774, PCT/AU00/00775, PCT/AU00/00776, PCT/AU00/00777,PCT/AU00/00770, PCT/AU00/00769, PCT/AU00/00771, PCT/AU00/00772,PCT/AU00/00754, PCT/AU00/00755, PCT/AU00/00756 and PCT/AU00/00757.

The disclosures of these co-pending applications are incorporated hereinby cross-reference.

Various methods, systems and apparatus relating to the present inventionare disclosed in the following co-pending applications filed by theapplicant or assignee of the present invention on 24 May 2000:

PCT/AU00/00518, PCT/AU00/00519, PCT/AU00/00520, PCT/AU00/00521,PCT/AU00/00522, PCT/AU00/00523, PCT/AU00/00524, PCT/AU00/00525,PCT/AU00/00526, PCT/AU00/00527, PCT/AU00/00528, PCT/AU00/00529,PCT/AU00/00530, PCT/AU00/00531, PCT/AU00/00532, PCT/AU00/00533,PCT/AU00/00534, PCT/AU00/00535, PCT/AU00/00536, PCT/AU00/00537,PCT/AU00/00538, PCT/AU00/00539, PCT/AU00/00540, PCT/AU00/00541,PCT/AU00/00542, PCT/AU00/00543, PCT/AU00/005.44, PCT/AU00/00545,PCT/AU00/00547, PCT/AU00/00546, PCT/AU00/00554, PCT/AU00/00556,PCT/AU00/00557, PCT/AU00/00558, PCT/AU00/00559, PCT/AU00/00560,PCT/AU00/00561, PCT/AU00/00562, PCT/AU00/00563, PCT/AU00/00564,PCT/AU00/00565, PCT/AU00/00566, PCT/AU00/00567, PCT/AU00/00568,PCT/AU00/00569, PCT/AU00/00570, PCT/AU00/00571, PCT/AU00/00572,PCT/AU00/00573, PCT/AU00/00574, PCT/AU00/00575, PCT/AU00/00576,PCT/AU00/00577, PCT/AU00/00578, PCT/AU00/00579, PCT/AU00/00581,PCT/AU00/00580, PCT/AU00/00582, PCT/AU00/00587, PCT/AU00/00588,PCT/AU00/00589, PCT/AU00/00583, PCT/AU00/00593, PCT/AU00/00590,PCT/AU00/00591, PCT/AU00/00592, PCT/AU00/00594, PCT/AU00/00595,PCT/AU00/00596, PCT/AU00/00597, PCT/AU00/00598, PCT/AU00/00516,PCT/AU00/00517 and PCT/AU00/00511.

The disclosures of these co-pending applications are incorporated hereinby cross-reference.

BACKGROUND

Various schemes have been proposed to add a degree of “interactivity” toa printed document in conjunction with a computer system. These includeprinting barcodes in the document which encode universal resourcelocators and thereby allow printed “hyperlinks” to be activated. To makethe interactivity of a printed document more seamless, and to supportmore sophisticated functions such as handwriting entry, it has beenproposed to code the surface of a document with position-indicating orfunction-indicating data which is effectively invisible to the unaidedhuman eye. The document typically includes data visible to the user, andthe user ostensibly interacts with this visible data using a sensingdevice which in reality detects and decodes the invisible coded data.

The coded data may be in the form of discrete tags, each of whichencodes data separately from other discrete tags. To decode the data indiscrete tags it is first necessary to identify the discrete tag and itsorientation relative to the sensing device. This usually requires theinclusion of target structures in the tag, which add to the overall sizeof each tag. Target structures may be shared between adjacent tags toreduce the effective overall size of each tag. One drawback of discretetags is the need for the sensing device to have at least one completetag in its field of view. Coupled with the possible need to allow thesensing device to be tilted with respect to the surface, the field ofview of the sensing device may need to be significantly larger than thetag size, as discussed in the present applicant's co-pending PCTApplication WO 00/72249.

Schemes have been proposed which use self-registering patterns andthereby dispense with explicit targets. When using a self-registeringpattern, the pattern of the data itself can implicitly supportoperations normally supported by explicit targets, includingdetermination of the orientation of the pattern with respect to thesensing device and determination of the alignment of the data within thepattern.

In its simplest form, a self-registering pattern consists of arectangular or other regular grid of glyphs. Each glyph is spatiallyseparated from its neighbors so that it can be distinguished from itsneighbors. This provides the first level of registration. Grid lines canthen be fitted through the points defined by the glyphs to identify theoriented (and possibly perspective-distorted) rectangular grid. Thisprovides the second level of registration, allowing glyphs to be sampledrelative to the grid. The self-registering pattern must contain a targetpattern which, once located, allows the orientation and translation ofthe glyph grid to be determined. This provides the third and final levelof registration, allowing glyph data to be assembled into data packetsand interpreted. So-called m-sequences, because of their maximal-lengthand cyclic properties, have been proposed as the basis for variousself-registering position-coding patterns.

The sensing and subsequent decoding of a position-coding pattern on asurface may be subject to error, due, for example, to the surface beingdamaged or dirty. Self-registering patterns do not directly support thedetection and/or correction of errors.

SUMMARY OF THE INVENTION

An arbitrary number of copies of a codeword of a q-ary cyclic (n, k)code C can be concatenated to form a sequence of arbitrary length. Awindow of size n onto the sequence is then guaranteed to yield acodeword of C. If the code is designed to contain exactly n codewords,then the “dimension” of the code is k=log_(q)n. If the code is designedso that all n codewords belong to the same and only cycle, then thewindow will yield n different codewords at n successive positions. Sincethere is a direct correspondence between a codeword and a position inthe, sequence (modulo n), each codeword can be uniquely mapped to one ofn (relative) positions. We refer to such a code as a cyclic positioncode. When the code is designed to have a minimum distance of d_(min),any number of errors up to d_(min)−1 can be detected, and any number oferrors up to └(d_(min)−1)/2┘ can be corrected.

Accordingly, the invention provides, in one broad form, a method ofdetermining a first offset, modulo n, of at least one point with respectto a first sequence of at least n symbols, the first sequence consistingof a repeating first codeword of a first cyclic position code, the firstcyclic position code having length n and minimum distance d_(min), themethod including:

-   -   obtaining, from the first sequence and at a position        corresponding to the at least one point, a first subsequence of        length w symbols, where w≧n−d_(min)+1;    -   mapping the first subsequence to a codeword of the first cyclic        position code; and    -   determining an offset, in the first sequence, of the codeword        thus obtained, and thereby determining the first offset.

Mapping the first subsequence preferably includes selecting a codewordof the first cyclic position code which matches the first subsequence.If there is no match between the first sub-sequence and a codeword thenan error may be flagged.

Alternatively, mapping the first subsequence includes selecting acodeword of the first cyclic position code most likely to match thefirst subsequence in the presence of up to └(d_(min)−1)/2┘ symbol errorsin the first subsequence. This may include selecting a codeword of thefirst cyclic position code closest in Hamming distance to the firstsubsequence. The Hamming distance is preferably defined over wcoordinates of the first cyclic position code.

The first sequence is preferably represented by a first pattern disposedor formed on or in a first substrate, and the method preferably includesobtaining the first subsequence by detecting or sensing at least part ofthe first pattern. Successive symbols of the first sequence arepreferably represented by successive parts of the first pattern arrangedin a substantially linear fashion.

To allow a minimum degree of error correction, it is preferable thatw≧n−d_(min)+2. To allow a greater degree of error correction, it ispreferable that w≧n.

The method may include obtaining additional subsequences from additionalsequences, and thereby additional offsets. The method may furtherinclude deriving differences between off-sets obtained from pairs ofadjacent sequences.

The invention also provides, in another broad form, a method ofdetermining a first coordinate value of at least one point with respectto a plurality of first sequences, each of the first sequencesconsisting of a repeating first codeword of a first cyclic positioncode, the first cyclic position code having length n₁ and minimumdistance d_(min) ₁ , the method including:

-   -   obtaining, from each of h₁ of the first sequences and at a        position corresponding to the at least one point, a respective        first subsequence of length w₁ symbols, where h₁≧2 and        w₁≧n₁−d_(min) ₁ +1;    -   mapping each of the first subsequences to a respective codeword        of the first cyclic position code;    -   determining an offset, in the corresponding first sequence, of        each codeword thus obtained, and thereby determining a        respective one of a plurality of first offsets of the at least        one point;    -   deriving, for each of h₁−1 pairs of the first sequences, a        difference between the corresponding pair of first offsets, and        thereby deriving a respective one of a plurality of first        differences; and    -   deriving, from the plurality of first differences, the first        coordinate value.

Preferably at least one of the first differences is interpreted as adigit of the first coordinate value. Preferably also, at least one ofthe first differences is interpreted as a marker separating the firstcoordinate value from an adjacent coordinate value.

This difference coding approach can also be used to encode both anexplicit position and a codeword, where the codeword is used for errordetection. The codeword may be encoded using one bit of each difference,for example. Such a codeword, if cyclic, can also be used to determineregistration of the position data, obviating the need for an explicitmarker difference. In general, a cyclic position code can be embedded in(or co-located with) other data to provide a registration signal forthat data.

One or more columns (and/or rows) per coordinate may be reserved forlocation-specific data.

The method preferably includes determining a second coordinate value ofthe at least one point with respect to a plurality of second sequences,each of the second sequences consisting of a repeating second codewordof a second cyclic position code, the second cyclic position code havinglength n₂ and minimum distance d_(min) ₂ , the method including:

-   -   obtaining, from each of h₂ of the second sequences and at a        position corresponding to the at least one point, a respective        second subsequence of length w₂ symbols, where h₂≧2 and        w₂≧n₂−d_(min) ₂ +1;    -   mapping each of the second subsequences to a respective codeword        of the second cyclic position code;    -   determining an offset, in the corresponding second sequence, of        each codeword thus obtained, and thereby determining a        respective one of a plurality of second offsets of the at least        one point; and    -   deriving, from the plurality of second offsets, the second        coordinate value.

Preferably, at least one of the second offsets is interpreted as a digitof the second coordinate value. Alternatively, the method furtherincludes deriving, for each of h₂−1 pairs of the second sequences, adifference between the corresponding pair of second offsets, and therebya respective one of a plurality of second differences; and includesderiving, from the plurality of second differences, the secondcoordinate value.

Preferably, each of the plurality of first sequences is represented by arespective one of a plurality of first patterns disposed or formed on orin a first substrate, the method including obtaining the correspondingfirst subsequence by detecting or sensing at least part of thecorresponding first pattern.

Preferably also, successive symbols of each of the first sequences arerepresented by successive parts of the corresponding first patternarranged in a substantially linear fashion, and the first patterns arearranged in a substantially parallel and spaced apart fashion.

The first and second coordinates may define orthogonal coordinates, forexample x and y coordinates in a Cartesian coordinate system. The firstand second cyclic position codes used to encode orthogonal coordinatesmay be of the same or different length, and may be the same or differentcodes.

Optimal binary (q=2) cyclic position codes of various lengths are givenin Table 3. The reverse, the complement and the reverse complement ofthe codes listed in the table are equally optimal. Many other cyclicposition codes also satisfy the criteria of the invention. Many optimalcodes are simplex codes.

The invention shall be better understood from the following,non-limiting, description of preferred embodiments of the invention withreference to the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows position decoding using a 7-bit cyclic position code;

FIG. 2 shows position decoding using a 7-bit cyclic position code in thepresence of errors;

FIG. 3 shows elements of difference coding using a 7-bit cyclic positioncode;

FIG. 4 shows an array of thirteen sequences of a 7-bit cyclic positioncode arranged in columns to enable difference coding of horizontalcoordinates;

FIG. 5 shows derivation of a first coordinate position from a first oneof the windows onto the array of FIG. 4;

FIG. 6 shows derivation of a second coordinate position from a secondone of the windows onto the array of FIG. 4;

FIG. 7 shows derivation of a second coordinate position from a third oneof the windows onto the array of FIG. 4;

FIG. 8 shows an interleaving of two orthogonal one-dimensional positioncodes;

FIG. 9 shows the interleaving of FIG. 8 represented using the presenceor absence of a single undifferentiated glyph;

FIG. 10 shows the interleaving of FIG. 8 represented using two distinctglyphs;

FIG. 11 shows an alternative interleaving of two orthogonalone-dimensional position codes;

FIG. 12 shows the interleaving of FIG. 11 represented using threedistinct glyphs and an empty glyph;

FIG. 13 shows the interleaving of FIG. 11 represented using a singleundifferentiated glyph offset in one of four directions from its nominalposition;

FIG. 14 shows a position-coding array having columns and rows oflocation-specific data;

FIG. 15 shows the mapping of data from location-specific columns androws in the array of FIG. 14 into tag-centered location-specificcodewords;

FIG. 16 shows the mapping of data from location-specific columns androws in the array of FIG. 14 into interstitial location-specificcodewords; and

FIG. 17 shows the codewords of FIGS. 15 and 16 at their correspondinglocations on the surface.

DESCRIPTION OF PREFERRED AND OTHER EMBODIMENTS

The embodiments of the invention utilize cyclic position codes in theencoding of positional and functional information on surfaces,principally in the form of self-registering patterns.

Position Coding Using M-Sequences

A linear feedback shift register (LFSR) of length k consists of k 1-bitstages numbered 0 to k−1. On each clock the content of stage 0 forms thenext bit of the output sequence, the content of stage i is moved tostage i−1, and the new content of stage k−1 is a feedback bit calculatedby adding together modulo 2 the previous contents of a fixed subset ofthe stages of the register (see Menezes, A. J., P. C. van Oorschot andS. A. Vanstone, Handbook of Applied Cryptography, 1997, CRC Press). Amaximum-length LFSR produces as output a so-called m-sequence with alength of 2^(k)−1, in which every possible non-zero register valueappears once before the sequence repeats. Because each k-bit valueappears exactly once in the m-sequence, a k-bit window into a knownm-sequence yields a unique k-bit subsequence which in turn can beinterpreted as a unique position within the m-sequence. Because of thecyclic nature of an m-sequence, a k-bit window onto a recurringm-sequence yields a locally unique position, i.e. modulo the length n ofthe m-sequence. Repeating or recurring m-sequences are also known aspseudo-noise (PN) sequences. The characteristics and construction of PNsequences are discussed extensively in Golomb, S. W., Shift RegisterSequences, Aegean Park Press, 1982, the contents of which are hereinincorporated by reference.

As described by F. J. MacWilliams and N. J. A. Sloane in “Pseudo-RandomSequences and Arrays” (Proceedings of the IEEE, Vol. 64, No. 12,December 1976, the contents of which are herein incorporated byreference), this windowing property of m-sequences can be extended totwo dimensions by folding an m-sequence into a two-dimensional array.The length of the m-sequence must be of the form n=2^(k) ¹ ^(k) ² −1,such that n₁=2^(k) ¹ −1 and n₂=n/n₁ are relatively prime and greaterthan 1. The output n₁×n₂-bit array is filled by writing the m-sequencedown the main diagonal of the array and continuing from the oppositeside whenever an edge is reached.

As described in PCT Application WO 92/17859 (J. Burns and S. Lloyd), thewindowing property can also be extended to two dimensions by arranging arecurring m-sequence into columns and encoding a second m-sequence intothe relative vertical alignment of adjacent columns. The columns areshifted cyclically to preserve the rectangular shape of the overallarray. Assuming the two m-sequences hove lengths 2^(k) ¹ −1 and 2^(k) ²−1 respectively, a k₁×(k₂+1) window into the (2^(k) ¹ −1)×2^(k) ² -bitarray yields a unique k₁×(k₂+1) subarray. Each k₁-bit column of thesubarray yields a relative k₁-bit position, and each pair of columnsyields a one-bit difference for a total of k₂ bits and hence a k₂-bithorizontal position. Once the horizontal position is known, thecumulative vertical shift associated with the horizontal position can beadded to the first column's relative vertical position to yield anabsolute vertical position.

By encoding k₁ bits rather than one bit into the relative verticalalignment of each pair of columns, i.e. by utilizing the full range ofrelative shifts allowed by the length of the vertical m-sequence, theposition-coding density of the array can be expanded to k₁×k₂horizontally. Each k₁-bit difference value then represents one elementof a k₁-ary m-sequence.

To recover k₁×k₂ bits in both dimensions, the difference coding approachused for the horizontal dimension can also be utilized for the verticaldimension, i.e. by arranging a recurring m-sequence into rows, andencoding an m-sequence into the relative alignment of adjacent rows. Them-sequence columns and rows must be spatially interleaved in a knownmanner to allow decoding. An m-sequence can recur any number of timeswithin a column (or row) to produce an overall pattern of sufficientsize.

As described in U.S. Pat. No. 6,208,771 (D. A. Jared et al), orthogonalcoordinates can be encoded using m-sequences placed in parallel to eachother. Each m-sequence is replicated every second row, witheven-numbered rows successively offset two bits to the left andodd-numbered rows successively offset two bits to the right.Even-numbered rows thereby define lines of constant coordinate at 45°,and odd-numbered rows define lines of constant coordinate at −45°, i.e.at 90° to each other; with each of their intersections thereby defininga two-dimensional position. Because the angles are induced by offsettingmultiple copies of the m-sequences, the approach is highly inefficient.

A drawback of any pure m-sequence approach is that as the requiredposition precision increases, it is increasingly expensive to translatea given subsequence (or subarray) into a position.

As described in PCT Application WO 00/73887 (P. Ericsson), thedifference coding approach can be used to code a coordinate directly,i.e. the relative vertical alignment of a set of recurring m-sequencecolumns can code adjacent parts of a horizontal position coordinate, andthe relative horizontal alignment of a set of recurring m-sequence rowscan code adjacent parts of a vertical position coordinate. Unlike a purem-sequence approach, a marker value (or range) must then be used toindicate which column (or row) separates the least-significant part ofone coordinate from the most-significant part of the next coordinate. Asan optimisation to the difference coding approach, only one coordinateis coded using the difference coding approach, while the othercoordinate is coded directly. This is possible since once thedifference-coded coordinate is decoded, the absolute alignment of therecurring m-sequences which code the orthogonal coordinate is known,rather than just their relative alignment. In the described approach,the two orthogonal sets of m-sequences are combined by pairing spatiallycoincident bits and representing each bit pair by a single glyph whichcan assume one of four possible values.

As described in PCT Application WO 92/17859 (J. Burns and S. Lloyd),so-called orientable m-sequences can be utilized in various ways toensure that the correct orientation of a subarray can be determined. Anorientable m-sequence is constructed so that if it contains a particularsubsequence then it does not contain the reverse of the subsequence.When imaging an array which requires a window size of k×k, a field ofview with a diameter of at least k√{square root over (2)} is requiredunless the orientation of the array with respect to the image sensor isconstrained.

Error Detection and Correction

Assume the data to be coded is broken into k-symbol blocks, with theq-ary symbols taken from the Galois field GF(q). The collection of allpossible k-tuples m=(m₀, m₁, . . . , m_(k−1)) forms a vector space overGF(q), containing q^(k) possible vectors. A corresponding block errorcode C of length n consists of a set of M n-symbol codewords {c₀, c₁, .. . , c_(M−1)}, where M=q^(k) and n>k, with each codeword of the formc=(c₀, c₁, . . . , c_(n−1)). Given a data block to be encoded, theencoder maps the data block onto a codeword in C. Since the collectionof all possible n-tuples over GF(q) contains q^(n) vectors, but thereare only M=q^(k) codewords, the code contains redundancy. This isexpressed logarithmically by r=n−log_(q)M=n−k, or by the code rateR=k/n. The code C is a linear code if it forms a vector subspace overGF(q), i.e. if it is closed under addition and under multiplication by ascalar (and thus contains the zero vector). The code is then said tohave dimension k and is referred to as an (n, k) code.

The Hamming distance between two codewords is the number of symbols inwhich the two codewords differ. The minimum distance d_(min) of a blockcode is the smallest Hamming distance of any pair of distinct codewordsin the code. The maximum distance d_(max) is the largest Hammingdistance of any pair of distinct codewords in the code.

An error pattern introduces symbol errors into a codeword. It ischaracterized by its weight, i.e. the number of symbols it corrupts. Foran error pattern to be undetectable, it must cause a codeword to looklike another codeword. A code with a minimum distance of d_(min) canthus detect all error patterns of weight less than or equal tod_(min)−1. Although a given code can detect many error patterns withgreater weights, this provides a limit on the weight for which a codecan detect all error patterns.

Given a sampled word possibly corrupted by an error pattern, the decodermaps the sampled word onto a codeword in C in such a way as to minimizethe probability that the codeword is different from the codewordoriginally written, and then maps the codeword onto a data block. In theabsence of a more specific characterization, it is assumed thatlower-weight error patterns are more likely than higher-weight errorpatterns, and that all error patterns of equal weight are equallylikely. The maximum likelihood written codeword is thus the codewordwhich is closest in Hamming distance to the sampled word. If the sampledword is closer to an incorrect codeword than the correct (written)codeword, then the decoder commits an error. Since codewords are bydefinition at least a distance d_(min) apart, decoder errors are onlypossible if the weight of the error pattern is greater than or equal tod_(min)/2. A maximum likelihood decoder can thus correct all errorpatterns of weight less than or equal to └(d_(min)−1)/2┘. Equivalently,the decoder can correct t errors so long as 2t<d_(min).

The minimum distance of a linear code is limited by the Singleton bound:d_(min)≦n−k+1. Codes which satisfy the Singleton bound with equality arecalled maximum-distance separable (MDS). Reed-Solomon codes (see Wicker,S. B. and V. K. Bhargava, eds., Reed-Solomon Codes and TheirApplications, IEEE Press, 1994, the contents of which are hereinincorporated by reference) are the most commonly-used MDS codes. Nobinary codes are MDS.

An erasure is a symbol of a sampled word assumed to have been corrupted.Since its position in the codeword is known, it can be ignored for thepurposes of decoding rather than being treated as an error. For example,the distance between the erased symbol in the sampled word and thecorresponding symbol in a codeword is not included in the Hammingdistance used as the basis for maximum likelihood decoding. Each erasurethus effectively reduces the minimum distance by one, i.e., in thepresence off erasures, up to └(d_(min)−f−1)/2┘ errors can be corrected.Equivalently, the decoder can correct t errors and f erasures so long as2t+f<d_(min). For an MDS code this becomes 2t+f<n−k+1.

A code is systematic if each of its codewords contains, withoutmodification, its corresponding data block at a fixed location. It isthen possible to distinguish between the data (or message) coordinatesof the code and the redundancy (or parity) coordinates of the code.

The rate of a linear code can be increased by puncturing the code, i.e.by deleting one or more of its redundancy coordinates. By the deletionof g coordinates, an (n, k) code is transformed into an (n−g, k) code.The minimum distance of the punctured code is d_(min)−g. Clearly, ifd_(min)−g<2, puncturing destroys the code's ability to correct even oneerror, while if d_(min)−g<1, it destroys the code's ability to detecteven one error. Equivalently, the length w=n−g of the punctured codemust obey w≧n−d_(min)+1 to be error-detecting, and w≧n−d_(min)+2 to beerror-correcting. The decoder for a punctured code can simply treatdeleted coordinates as erasures with respect to the original code.

A block code C is a cyclic code if for every codeword c=(c₀, c₁, . . . ,c_(n−2), c_(n−1))εC, there is also a codeword c′=(c_(n−1), c₀, c₁, . . ., c_(n−2))εC, i.e. c′ is a right cyclic shift of c. It follows that alln cyclic shifts of c are also codewords in C. If the number of codewordsq^(k) exceeds the length of the code n, then the code contains a numberof distinct cycles, with each cycle i containing s_(i) unique codewords,where s_(i) divides n. If the code contains the zero vector, then thezero vector forms its own cycle.

Cyclic Position Codes

Position decoding via a k-symbol window onto a recurring m-sequence oflength n=2^(k)−1 does not allow error detection or correction. However,position decoding via an n-symbol window onto a recurring cycliccodeword of length n does allow error detection and correction.

An arbitrary number of copies of a codeword of a cyclic (n, k) code Ccan be concatenated to form a sequence of arbitrary length. A window ofsize n onto the sequence is then guaranteed to yield a codeword of C. Ifthe code is designed to contain exactly n codewords, then the dimensionof the code is, by definition, k=log_(q)n. If the code is designed sothat all n codewords belong to the same and only cycle, then the windowwill yield n different codewords at n successive positions. Since thereis a direct correspondence between a codeword and a position in thesequence (modulo n), each codeword can be uniquely mapped to one of n(relative) positions. Significantly, a position can be determined evenin the presence of up to └(d_(min)−1)/2┘ symbol errors. We refer to sucha code as a cyclic position code. Any codeword of a cyclic position codedefines the code. A cyclic position code is not a linear code, since itdoes not contain the zero vector. However, it is useful to use theterminology of linear codes in the following discussion andcharacterisation of cyclic position codes. Many good cyclic positioncodes are linear codes with the zero vector removed.

Note that the cyclic shifts of a binary m-sequence of length 2^(k)−1constitute all of the non-zero codewords of a linear cyclic code oflength n=2^(k)−1, dimension k, and minimum distance d_(min)=2^(k)−1 (seeF. J. MacWilliams and N. J. A. Sloane, “Pseudo-Random Sequences andArrays”, Proceedings of the IEEE, Vol.64, No.12, December 1976). Abinary m-sequence therefore defines a cyclic position code. Note also,however, that the m-sequence-based position coding schemes describedearlier do not use m-sequences as codewords, since they use a windowsize of k rather than a window size of n.

More generally, m-sequences define a subset of the set of simplex codes.The simplex codes have length n=4m−1 and minimum distanced_(min)=(n+1)/2=2m. As implied by the name, the codewords of a simplexcode define the equidistant vertices of an n-simplex. The minimum andmaximum distances of a simplex code are therefore the same. For n prime,the Paley construction can be used to construct a cyclic simplex codeusing quadratic residues (see MacWilliams, F. J. and N. J. A. Sloane,The Theory of Error-Correcting Codes, North-Holland, 1977, and Wicker,S. B., Error Control Systems for Digital Communication and Storage,Prentice Hall, 1995, the contents of both of which are hereinincorporated by reference). For n prime or n=2^(k)−1, then, a simplexcode is cyclic and therefore defines a cyclic position code.

A cyclic simplex code of length n=4 m−1 defines an optimal cyclicposition code in the sense that it has the largest minimum distancepossible not only for its length but for any length n<4(m+1)−1.

The “dimension” of a cyclic position code is fractional unless thelength of the code is an integer power of the symbol size. We consider acyclic position code “systematic” if all of its codewords are distinctin k′=┌log_(q)n┐ symbols. An m-sequence defines a systematic cyclicposition code, whereas a cyclic simplex code in general may not.

When a cyclic position code is punctured, one or more symbols aresystematically deleted from each codeword of the code. However, since apunctured cyclic position code is not cyclic, the original codeword isstill used to construct the arbitrary length code sequence on whichpunctured position decoding is based. If the punctured code has a lengthof n−g, then an (n−g)-symbol window is used onto a recurring cycliccodeword of length n from the original code.

Cyclic Position Code Example

By way of example, Table 1 shows the 7 codewords of a 7-bit binarycyclic position code. The code is a cyclic simplex code and issystematic. It has a minimum (and maximum) distance of 4 and thereforeallows one error to be corrected.

TABLE 1 codeword shift 0001011 0 0010110 1 0101100 2 1011000 3 0110001 41100010 5 1000101 6

Table 2 shows the Hamming distance between the first codeword of thecode and each of the other codewords of the code, computed with aone-bit error successively in each of the seven possible locations inthe codeword. In each case the corrupted (i.e. inverted) bit isindicated by ♦. Whereas the distance between the corrupted codeword andits uncorrupted original is exactly one in each case, the distancebetween the corrupted codeword and each of the other codewords is neverless than three. Since the code (by definition) only contains a singlecycle, the table demonstrates the ability of the code to correct anyone-bit error in any codeword.

TABLE 2 codeword 0001011 0010110 0101100 1011000 0110001 1100010 1000101000101♦ 1 3 3 3 5 3 5 00010♦1 1 5 3 3 3 5 3 0001♦11 1 3 3 5 5 5 3000♦011 1 3 5 5 3 3 3 00♦1011 1 3 5 3 3 5 5 0♦01011 1 5 3 5 3 3 5♦001011 1 5 5 3 5 3 3

FIG. 1 shows a sequence 50 consisting of the first codeword of the 7-bitbinary cyclic position code repeated multiple times. FIG. 1 also showseight adjacent 7-bit windows 52 onto the sequence, each yielding acodeword of the code and thereby a shift value 54 according to Table 1.FIG. 2 shows the same sequence 50 in the presence of two one-bit errors60 and 62 respectively. Seven of the eight 7-bit windows 52 onto thesequence therefore also contain one-bit errors. These errors arecorrectable as illustrated in Table 2, yielding corresponding validcodewords 58 and thereby shift values 54 as before.

Optimal Cyclic Position Codes

For modest code lengths and symbol sizes, an optimal cyclic positioncode of a particular length and symbol size can be found by exhaustivesearch. Table 2 lists the characteristics of optimal binary cyclicposition codes of various lengths, together with specific examples.Although in most cases the optimal code is systematic, for lengths of 11and 12 a non-systematic code is optimal. Note that the optimal codes oflength 7 and 15, and generally of length n=2^(k)−1, are defined bym-sequences, as discussed earlier.

TABLE 3 n k d_(min) d_(max) t dr_(min) code 7 3 4 4 1 2 0001011 8 3 4 61 2 00010111 9 4 4 6 1 2 000010011 000010111 000100111 10 4 4 6 1 20000100111 8 0000101111 6 0000110111 0001010011 11 4 4 8 1 2 000010011116 00001010011 8 00010100111  5^(a) 6 6 2 4 00010010111 12 4 4 8 1 4000100110111  5^(a) 6 8 2 4 000010110111 13 4 6 8 2 4 0000100110111 14 46 10 2 4 00001001101111 00010011010111 15 4 8 8 3 4 000010100110111 16 48 10 3 4 0000100110101111 0000101100111101 17 5 8 12 3 400000100011010111 10, 12 . . . 18 5 8 12 3 6 000001100101101111000010011011110101 14 000010101101001111 12 000010101111001101 19 5 1010 4 8 0000101011110010011 20 5 10 14 4 6 00000100101011001111 1200000101011110010011 00000101011110011011 00001010111011001111 21 5 1012 4 8 000001010111100100011 22 5 10 14 4 8 00000100011010100111110000010011000111101101 0000010011011000111101 0000011001110110101111 235 10 14 4 8 00000100011001110101111 14, 16 . . .  7^(a) 12 12 5 1000000101001100110101111 24 5 12 16 5 8 000010010110111010001111 25 5 1216 5 8 0000010001100101011011111 14, 16, 18 . . . 26 5 12 16 5 1000000100011010100111110111 27 5 12 16 5 10 000001100101101010001001111000001101100111101001010111 28  6^(a) 14 16 6 100000001001010111010011001111 0000001011011001010111100011 29 5 14 10 1800000100101011111001100011101 00000100101011111010001100111  13^(a)  1412 6 16 00000100111110010101100101011  8^(a)00001110001000100101101110111 31  7^(a) 16 16 70001001000011101010001111011011 43  7^(a) 22 22 10 200011010110001000001110100011111011100101001 47  9^(a) 24 24 11 2200000100001101010001101100100111010100111101111 59  9^(a) 30 30 14 28001000101011011000100001100000111110011110111001 00101011101 ^(a)Notsystematic.

For coding purposes, a code is considered equivalent to its reverse, itscomplement, and its reverse complement, so only one of these is includedin the table in each case.

Although the table (mostly) lists systematic cyclic position codes, forcode lengths greater than 7 there are many more non-systematic codeswhich are also optimal (indicated by ellipsis).

A particular cyclic position code can be punctured to reduce its lengthand hence its position-coding precision (and corresponding window size).Table 2 shows that the punctured code is superior to any other code ofthe same length when the code being punctured is a cyclic simplex code,since each cyclic simplex code in the table has a minimum distancegreater by two than its predecessor in the table.

For example, the optimal (cyclic simplex) code of length 19 in thetable, when punctured to a length of 18, is superior to the four optimalcodes of length 18 in the table. The optimal length-18 codes have aminimum distance and minimum reverse distance of 8 and 6 respectively,while the punctured length 19 code has a superior minimum distance andminimum reverse distance of 9 and 7 respectively.

The first column of Table 4 shows the 19 codewords of the optimallength-19 code in Table 3. The second column of Table 4 shows the 19codewords of an optimal length-18 code obtained by puncturing thelength-19 code.

TABLE 4 corresponding original (length-19) punctured (length-18)codeword codeword 0000101011110010011 0000101011110010010001010111100100110 000101011110010011 0010101111001001100001010111100100110 0101011110010011000 0101011110010011001010111100100110000 101011110010011000 0101111001001100001010111100100110000 1011110010011000010 1011110010011000010111100100110000101 011110010011000010 1111001001100001010111100100110000101 1110010011000010101 1110010011000010101100100110000101011 110010011000010101 1001001100001010111100100110000101011 0010011000010101110 0010011000010101110100110000101011110 010011000010101111 1001100001010111100100110000101011110 0011000010101111001 0011000010101111000110000101011110010 011000010101111001 1100001010111100100110000101011110010 1000010101111001001 100001010111100100

One design goal which might favor a particular non-systematic code isthe maximization of the number of set bits in the codewords. This isrelevant if the presence or absence of an undifferentiated glyph is usedto represent each bit, since it can maximize our ability to discern thestructure of the overall glyph array.

Since a cyclic position code of length n contains only n codewords, formodest code lengths it is tractable to decode a sampled word bycalculating the Hamming distance between the sampled word and eachcodeword in turn, and then choosing the closest codeword. For very shortcodes a lookup table can also be used. Algebraic decoding can be usedfor longer codes (see Wicker, S. B., Error Control Systems for DigitalCommunication and Storage, Prentice Hall, 1995, and Berlekamp, E. R.,Algebraic Coding Theory, Aegean Park Press, 1984, the contents of bothof which are herein incorporated by reference). If the sampled word isequidistant from two or more codewords, then it is preferable thatdecoding should report an error. Encoding can similarly be performeddirectly (e.g. by lookup) or algebraically.

The search for an optimal cyclic position code of length n and symbolsize q can proceed by generating all possible q^(n) codewords in turn,and for each codeword defining a code which contains the n cyclic shiftsof the codeword. The minimum distance and minimum reverse distance ofthe code are then calculated. If the code is the best found so far thenit is recorded. All optimal codes can be found in two passes, i.e. afirst pass to determine the characteristics of the optimal code, and thesecond pass to enumerate the codes which exhibit the optimalcharacteristics.

If the optimal cyclic position code is a cyclic simplex code then acorresponding construction technique can be used, i.e. the Paleyconstruction for n prime, or LFSR construction for n=2^(k)−1. Note thatfor some k (e.g. 5), the length n=2^(k)−1 (e.g. 31) is prime, and thePaley construction can be used.

Q-ary Cyclic Position Codes

A q-ary cyclic position code of length n=q can be constructed byconcatenating q distinct symbols into a codeword. The minimum (andmaximum) distance of the code is equal to its length, i.e.d_(min)=d_(max)=n. By way of example, Table 5 shows the 7 codewords of a7-symbol 7-ary cyclic position code with minimum distance 7. Whenn=q=2^(m)−1, the cyclic position code is equivalent to an (n, 1)Reed-Solomon code with the zero vector removed code in Table 5corresponds to a (7, 1) Reed-Solomon code, for example. Whereas a q-arycyclic position code may contain the zero symbol, the non-zero codewordsof an (n, 1) Reed-Solomon code won't contain the zero symbol, since thezero symbol only appears in the zero vector. A Reed-Solomon code willalso typically have a different symbol ordering.

TABLE 5 codeword shift 1234567 0 2345671 1 3456712 2 4567123 3 5671234 46712345 5 7123456 6

A q-ary cyclic position code of length n=q can be punctured to obtain aq-ary cyclic position code of arbitrary length n_(p)<q. No matter whatthe punctured length, d_(min)=n_(p).

A q-ary position code of length n>q can be designed, or derived from onecycle of a q-ary cyclic code such as a Reed-Solomon code (where k>1).When n>q, d_(min)<n. For MDS codes such as Reed-Solomon codes, forexample, d_(min)=n−k+1.

Difference Coding using a Cyclic Position Code

The various m-sequence-based position coding approaches describedearlier can be realized using a recurring codeword from a cyclicposition code rather than a repeating m-sequence, at the expense ofsampling more than k symbols to support error detection and errorcorrection.

By way of example, FIG. 3 shows the seven ways 70 adjacent sequences ofthe 7-bit code can be aligned. These provide the basic elements fordifference coding. The differences 72 in vertical position are shownbelow each pair of columns.

FIG. 4 shows fourteen (14) adjacent sequences 100 of the 7-bit codearranged into columns to encode, as base-6 differences, the digits oftwo successive horizontal coordinates, these being 10312₆ and 10313₆, asindicated at 102 and 104 respectively. The thin rectangles shown in thefigure are included to enable the reader to easily discern each instanceof the reference codeword. In practice only the bit values are encoded.As can be seen in the figure, the codeword 0001011 is repeated end onend in each column. A difference of 6 is used as a marker value (denotedby • in the figures) to separate the least-significant digit of onecoordinate from the most-significant digit of the next coordinate.

FIG. 4 also shows three 7×7 windows 106, 107 and 108 onto the set ofcolumns. FIGS. 5, 6 and 7 show the contents of the windows 106, 107 and108 respectively. Each 7-bit column within each window can beinterpreted as a codeword of the 7-bit code, in turn yielding a shiftvalue and hence a relative position according to Table 1. Since asampled 7-bit codeword may contain errors, it is first decoded to yielda valid codeword, e.g. using a maximum-likelihood decoder as describedearlier (i.e. either directly or algebraically). In FIGS. 4 to 7 anumber of sampled bits within each window are indicated, by way ofexample, as errors. Each bit denoted by ♦ indicates an error, i.e. thecorresponding bit has been inverted. Since in the example there is atmost one corrupted bit within a given codeword, each such error is fullycorrectable (as described earlier). After decoding each codeword, thedifference in shift values between each pair of adjacent columns thengives a digit of a coordinate value. The difference between any twoadjacent shift values, modulo 7, is invariant of the vertical positionof the window, since the offset of the codewords in each column does notchange in the vertical direction.

Horizontally each window yields 5 digits from the same coordinate orfrom two adjacent coordinates, as well as exactly one marker.

As seen in FIG. 5, window 106 yields shift values 110 of {1, 1, 5, 4, 2,3, 2} for the seven columns sampled, when sampled from the topdownwards. These in turn yield difference values 112 of {0,3, 1,2,6, 1}.

It is a simple matter to assemble a single coordinate even if the digitsspan two adjacent coordinates, as is the case with both windows 106 and107. The most-significant digits to the right of the marker are simplyprepended to the least-significant digits to the left of the marker.

In the example shown in FIGS. 4 to 7, and as mentioned earlier, thedifference of 6 is used as the marker (denoted by • in the figures).Thus the first window's sequence of difference values 112 {0, 3, 1, 2,6, 1} is treated as two subsequences {0, 3, 1, 2} and {1}, separated bythe marker 6, and hence as two coordinate fragments 0312₆ and 1₆. Theseare converted to a coordinate value 114 of 10312₆ by multiplying thesecond fragment by the precision of the first fragment (i.e. 10000₆, or6⁴ where 4 is the number of digits in the first fragment) and adding thefragments together. If adding 1 to the least-significant fragment wouldresult in carry, then 1 is subtracted from the most-significantfragment.

The second window 107 yields shift values of {0, 6, 4, 5, 4, 4, 1} whichin turn yield difference values of {1, 2, 6, 1, 0, 3} and a coordinatevalue of 10312₆ also. The third window 108 yields shift values of {3, 4,3, 3, 0, 6, 3} which yield difference values of {1, 0, 3, 1, 3, 6} and acoordinate value of 10313₆.

The position of the marker in the window can be used to generate ahigher-precision coordinate value, since it reflects the relativealignment of the window with respect to the coordinate. As shown inFIGS. 5 to 7, the length of the subsequence to the right of the markeris used to generate an additional fractional digit, i.e. when the markeris adjacent to the right-hand edge of the window, the window is definedto be aligned with the coordinate. Other nominal alignments are ofcourse possible. In the first window 106 the second subsequence haslength 1 and the fractional coordinate value 116 is thus 0.1₆. In thesecond window 107 the second subsequence has length 3 and the fractionalcoordinate value is thus 0.3₆. In the third window 108 the secondsubsequence has length 0 and the fractional coordinate value is thus0.0₆. Still higher-precision coordinate values can be generated from therelative position of the sensed coded data in the field of view of thesensing device, the perspective transform of the sensed coded data, andthe known geometry of the sensing device. This is described in moredetail in the present applicant's co-pending PCT Application WO00/72287.

The marker may be any difference value, although a difference of n−1 isparticularly convenient since it leaves a contiguous range of differencevalues 0 through n−2, each of which can be directly mapped to a base n−1digit. In general, of course, any predetermined mapping from differencevalues to coordinate value can be utilized.

Spatial Arrangement

A two-dimensional position coding array consists of two orthogonalone-dimensional position coding arrays spatially combined orinterleaved. A three-dimensional position coding array consists of threeorthogonal one-dimensional position coding arrays spatially combined orinterleaved. And so on for higher dimensions. A two-dimensional positioncoding array may be encoded on a surface. A three-dimensional positioncoding array may be encoded in a volume.

A one-dimensional position coding sequence may contain q-ary symbols,where q≧2. It is advantageous to use q distinct glyphs to represent theq distinct symbol values, the avoid the need to determine registrationat the symbol level. Conversely, to reduce the number of requiredglyphs, q may be minimized. In the presence of spatially-coherent(burst) noise, however, a larger symbol size provides more efficienterror correction.

To avoid the need to distinguish symbols belonging to orthogonalsequences, symbols may be paired and concatenated at each point ofintersection between two orthogonal position coding sequences, andrepresented by one of a set of q² glyphs.

In the approach of PCT Application WO 92/17859 (J. Burns and S. Lloyd),the two orthogonal one-dimensional arrays are spatially interleaved andare represented by different color pairs to allow subsequent separation.Orthogonal sequences can also be distinguished by utilizing differentbase sequences.

In the approach of PCT Application WO 00/73887 (P. Ericsson), the twoorthogonal one-dimensional arrays are combined by concatenatingspatially coincident bits and representing each bit pair by a singleglyph which can assume one of four possible values. The four possiblevalues are represented by a dot in one of four positions relative to anominal grid position. The alignment of the grid itself is determined byfitting straight lines through the off-grid dots.

FIG. 8 shows a possible spatial interleaving 118 of two orthogonalbinary one-dimensional position arrays 120 and 122. To avoid the need todistinguish different glyphs, the presence and absence of singleundifferentiated glyph can be used in place of two explicit glyphs, asshown in FIG. 9. Alternatively, two distinct glyphs can be used, asshown in FIG. 10.

FIG. 11 shows an alternative spatial interleaving 124 of two orthogonalbinary one-dimensional position arrays 126 and 128, where spatiallycoincident symbol values from the two arrays have been combined. FIG. 9shows the interleaving represented using three distinct glyphs and anempty glyph. FIG. 10 shows the interleaving represented using a singleundifferentiated glyph offset in one of four directions from its nominalposition (as suggested in PCT Application WO 00173887).

Assuming the structure of the overall glyph array can be discerned, itcan be partitioned into its two constituent orthogonal one-dimensionalposition coding subarrays. Each subarray of glyphs can be assigned to aset of one-dimensional position code sequences in four ways, i.e.corresponding to the four possible orientations of the subarray. Sincethe one-dimensional position code contains redundancy, it can bedesigned so that a correct assignment generates fewer errors than anincorrect assignment, even in the presence of errors due to otherfactors. Once the orientation of one subarray is known, the orientationof the other subarray follows. Alternatively, errors can be minimizedacross both subarrays to choose a correct orientation.

The number of errors resulting from certain incorrect assignments of asubarray can be quantified. To do so we define the reverse minimumdistance of a one-dimensional cyclic position code.

The reverse of a code is a code containing the symbol-wise reverse ofeach of the codewords of the code. The minimum reverse distance dr_(min)of a code is the smallest Hamming distance between a codeword in thecode and a codeword in the reverse of the code.

In the absence of errors due to other factors, the number of errorsresulting from a correct assignment of a subarray is zero. The number oferrors resulting from incorrect an assignment due to incorrect rotationof 180° has a lower bound of n×dr_(min).

Accumulated Shift

A two-dimensional position coding array is typically generated inisolated fragments, with each fragment coding the coordinate ranges of asingle surface such as a physical page. For proper continuity betweenfragments, the shift of the initial row and column in each fragmentshould reflect the accumulated shift associated with all previous rowsand columns. However, since continuity between fragments is not strictlyrequired, the first row and column of a fragment can have zero shifts.

The following discussion assumes continuity between fragments isdesired. For clarity it deals with the coding of an x coordinate usingdifferences between columns. The same approach applies to the coding ofa y coordinate using differences between rows.

For a given x coordinate, each difference between a pair of earliercolumns contributes to the accumulated shift and thus to the shift ofthe first column which encodes the coordinate. Assume each pair ofcolumns encodes a difference between zero and b−1. Further assume thatthe x coordinate x is represented by p base b digits a_(i), each encodedby a difference, such that:

$\begin{matrix}{x = {\prod\limits_{i = 0}^{p - 1}{a_{i}b^{i}}}} & \left( {{EQ}\mspace{14mu} 1} \right)\end{matrix}$

Assume that adjacent coordinates are separated by a marker “digit” withvalue v.

The accumulated shift B(b, i) due to b^(i) is given by:

$\begin{matrix}{{B\left( {b,i} \right)} = {{i\; b^{i - 1}{\sum\limits_{j = 0}^{b - 1}j}} = {{i\; b^{i - 1}{{b\left( {b - 1} \right)}/2}} = {i\;{{b^{i}\left( {b - 1} \right)}/2}}}}} & \left( {{EQ}\mspace{14mu} 2} \right)\end{matrix}$

The accumulated shift A(a_(i), b, i) due to a_(i)b^(i) is given by:

$\begin{matrix}{{A\left( {a_{i},b,i} \right)} = {{{a_{i}{B\left( {b,i} \right)}} + {b^{i}{\sum\limits_{j = 0}^{a_{i} - 1}j}}} = {{a_{i}{B\left( {b,i} \right)}} + {b^{i}{{a_{i}\left( {a_{i} - 1} \right)}/2}}}}} & \left( {{EQ}\mspace{14mu} 3} \right)\end{matrix}$

The accumulated shift up to but not including x is given by:

$\begin{matrix}\left. {{x\; v} + {\sum\limits_{i = 0}^{p - 1}\left( {A\left( {a_{i},b,i} \right)} \right)} + {a_{i}\left( {x\mspace{14mu}{mod}\mspace{14mu} b^{i}} \right)}} \right) & \left( {{EQ}\mspace{14mu} 4} \right)\end{matrix}$

These functions for efficiently computing accumulated shift can beconveniently implemented in a device which prints position-codedsurfaces, such as described in the present applicant's co-pending PCIApplications WO 00/72126 and WO 00/72127.

Position Coding Variations

As discussed earlier, a difference coding approach can be used to encodea position explicitly or via an m-sequence. It can also be used toencode a position as a codeword to allow error detection and errorcorrection.

A difference coding approach can also encode both an explicit positionand a codeword, where the codeword is used for error detection. Forexample, the codeword may be encoded using one bit of each difference.Such a codeword, if cyclic, can also be used to determine registrationof the position data, obviating the need for an explicit markerdifference.

In general, a cyclic position code can be embedded in (or co-locatedwith) other data to provide a registration signal for that data.

Embedded Function Flags

A difference coding approach per se does not make it easy to embedlocation-specific data, such as function flags (as discussed in thepresent applicant's co-pending PCT Application WO 01/41055), in thetwo-dimensional position coding array, since the two-dimensionalposition coding array defines two-dimensional positions through theinteraction of two orthogonal one-dimensional position coding arrays.

However, one or more columns (and rows) per coordinate may be reservedfor location-specific data. In this case the difference coding approachmust ignore those columns (and rows) for difference coding purposes.Conversely, the location-specific data must not be allowed to inducefalse registration, e.g. by impersonating the marker difference.

If the differences between a set of columns (or rows) encode a codewordfor the purposes of error detection and registration, as discussed inthe previous section, then the codeword provides sufficient informationfor data columns to be ignored. However, since the data columns maycontain arbitrary data, they may induce errors in the codeword. If thecodeword is error-correctable and, during error correction, the onlysymbols in error are found to be associated with the data columns, thenall is well. If errors also lie elsewhere, then the position data itselfis suspect, and the decoder should report an error.

If only a few bits are required for encoding location-specific data,then the adjacent bits can be chosen in such a way as to avoid inducinga marker difference in relation to adjacent columns. This needs to takeinto account that the data will be error-corrected before beinginterpreted. This approach can obviate the need for a registrationcodeword.

To allow errors in location-specific data to be detected and possiblycorrected, redundancy must be introduced. Since a continuous redundancyscheme based on a cyclic code doesn't support arbitrary data,location-specific data is best arranged into codewords of such a sizethat the sampling window is guaranteed to contain at least one completecodeword. If the sampling window has a size of n×n (i.e. to work inconjunction with a pair of orthogonal cyclic position codes of lengthn), then the size n′ of the location-specific data codeword must obeyn′≦n/2.

Rather than having two independent codewords of length n′ in the twoorthogonal one-dimensional position coding arrays, corresponding datafrom the two arrays can be combined to form a single codeword of up tolength n. The burst error detection and correction capability of thesingle larger codeword is better than that of the two smaller codewords,but at the added cost of sampling a larger area, as discussed below.

FIG. 14 shows a portion 130 of a two-dimensional position coding arraywhich encodes four two-dimensional positions, i.e. corresponding to four“tags” 132, 134, 136 & 138 shown with dashed outlines. It also shows,overlaid on the tags, columns 140, 142 and rows 144, 146, which are usedto encode location-specific data rather than position-codingdifferences.

Each tag-height portion of a data column (or tag-width portion of a datarow) is shown broken into four quarters, indicated by the solid squares148. Referring to FIG. 15, where a row and column intersect in thecenter of a tag it is natural to combine the corresponding data quartersinto a single codeword 150. Elsewhere it is natural to systematicallycombine four quarters 152, 154, 156 & 158 from four different rows andcolumns into a codeword 160 corresponding to an interstitial location.This is shown in FIG. 16, with the arrows indicating the source dataquarter corresponding to each interstitial data quarter. The result isthat location-specific data is coded at a frequency of √{square rootover (2)} times the tag frequency. FIG. 17 shows only the codewords 150and 160 so derived shown in the positions to which they correspond.

Using this scheme, location-specific data can be only be recovered at anarbitrary position if the size of the sampling window is expanded to3n/2, assuming the size of the combined location-specific data codewordis n.

Non-Cyclic Self-Registering Patterns

Whereas the previous section considered the use of subsequences ofcyclic sequences as target patterns, it is also conceivable to use anon-cyclic pattern as a target pattern. It is generally not possible touse a pure data portion of the overall pattern as a target pattern,since, as explained below, the probability of a false match isinsufficiently low.

We assume for the purposes of the following analysis that a Reed-Solomoncode is used to produce an error-correctable encoding of the data.

An (n, k) Reed-Solomon code is characterized by its length n anddimension k. The symbol size q of the code is given by:q=log ₂(n+1)  (EQ 5)

The maximum number of correctable errors t is given by:

$\begin{matrix}{t = \left\lfloor \frac{n - k}{2} \right\rfloor} & \left( {{EQ}\mspace{14mu} 6} \right)\end{matrix}$

For a particular codeword, the number of “aliases” which can be decodedcorrectly, i.e. which contain no more than t symbols in error, is givenby:

$\begin{matrix}{1 + {\sum\limits_{i = 1}^{t}{{{}_{\;}^{}{}_{}^{\;}}\left( {2^{q} - 1} \right)}^{i}}} & \left( {{EQ}\mspace{14mu} 7} \right)\end{matrix}$

This is approximated (and bounded) by:^(n)C_(t)2^(qt)  (EQ 8)

The number of distinct codewords containing no errors is given by:2^(qk)  (EQ 9)

From (EQ 8) and (EQ 9), the number of valid codewords is approximatedby:^(n)C_(t)2^(q(k+t))  (EQ 10)

The total number of codewords is given by:2^(qn)  (EQ 11)

From (EQ 10) and (EQ 11), the probability P of a false match istherefore approximated by:

$\begin{matrix}{P = \frac{{{}_{\;}^{}{}_{}^{\;}}2^{q{({k + t})}}}{2^{q\; n}}} & \left( {{EQ}\mspace{14mu} 12} \right)\end{matrix}$

From (EQ 6) this simplifies to:

$\begin{matrix}{P = \frac{{}_{\;}^{}{}_{}^{\;}}{2^{q\; t}}} & \left( {{EQ}\mspace{14mu} 13} \right)\end{matrix}$

From (EQ 13), for a (15, 5) code, P has an approximate upper bound of1/256. Using (EQ 7) as a more accurate basis for (EQ 10) and hence (EQ13), P is approximately 1/341.

The probability of a false match when a data codeword of arbitrary valueis used as the target pattern is therefore insufficiently low.

From (EQ 8) and (EQ 11), the probability Q of a false match when atarget pattern of specific value is used is approximated by:

$\begin{matrix}{Q = \frac{{{}_{\;}^{}{}_{}^{\;}}2^{q\; t}}{2^{q\; n}}} & \left( {{EQ}\mspace{14mu} 14} \right) \\{Q = \frac{{}_{\;}^{}{}_{}^{\;}}{2^{q{({n - t})}}}} & \left( {{EQ}\mspace{14mu} 15} \right)\end{matrix}$

From (EQ 15), for a (15, 5) code Q has an approximate upper bound of1/250,000,000, which may be acceptable in many applications.

It will be apparent to those skilled in the art that many obviousmodifications and variations may be made to the embodiments describedherein without departing from the spirit or scope of the invention.

1. A method of determining a first offset, modulo n, of at least onepoint with respect to a first sequence of at least n symbols, the firstsequence consisting of a repeating first codeword of a first cyclicposition code, the first cyclic position code having length n andminimum distance d_(min), the method including: obtaining, from thefirst sequence and at a position corresponding to the at least onepoint, a first subsequence of length w symbols, wherein w≧n−d_(min)+1;mapping the first subsequence to a codeword of the first cyclic positioncode; and determining an offset, in the first sequence, of the codewordthus obtained, and thereby determining the first offset.
 2. The methodof claim 1 wherein mapping the first subsequence includes selecting acodeword of the first cyclic position code which matches the firstsubsequence.
 3. The method of claim 1 wherein mapping the firstsubsequence includes selecting a codeword of the first cyclic positioncode most likely to match the first subsequence in the presence of up to└(d_(min)−1)/2┘ symbol errors in the first subsequence.
 4. The method ofclaim 1 wherein if there is no match between the first subsequence and acodeword an error is flagged.
 5. The method of claim 1 wherein mappingthe first subsequence includes selecting a codeword of the first cyclicposition code closest in Hamming distance to the first subsequence. 6.The method of claim 5 wherein the Hamming distance is defined over wcoordinates of the first cyclic position code.
 7. The method of claim 1wherein the first sequence is represented by a first pattern disposed orformed on or in a first substrate, the method including obtaining thefirst subsequence by detecting or sensing at least part of the firstpattern.
 8. The method of claim 7 wherein successive symbols of thefirst sequence are represented by successive parts of the first patternarranged in a substantially linear fashion.
 9. The method of claim 1wherein the first cyclic position code is a simplex code.
 10. The methodof claim 1 wherein the first cyclic position code is a code listed inTable 3, or is the reverse, the complement, or the reverse complement ofa code listed in Table
 3. 11. The method of claim 1 whereinw≧n−d_(min)+2.
 12. The method of claim 1 wherein w≧n.
 13. The method ofclaim 1 further including determining a second offset, modulo n, of theat least one point with respect to a second sequence of at least nsymbols, the second sequence consisting of the repeating first codewordof the first cyclic position code, the method including: obtaining, fromthe second sequence and at a position corresponding to the at least onepoint, a second subsequence of length w symbols; mapping the secondsubsequence to a codeword of the first cyclic position code; anddetermining an offset, in the second sequence, of the codeword thusobtained, and thereby determining the second offset.
 14. The method ofclaim 13 further including deriving a first difference between the firstoffset and the second offset.
 15. The method of claim 14 wherein thesecond sequence is represented by a second pattern disposed or formed onor in the first substrate, the method including obtaining the secondsubsequence by detecting or sensing at least part of the second pattern.16. The method of claim 13 further including determining a plurality ofadditional offsets, modulo n, of the at least one point, each withrespect to one of a plurality of additional sequences of at least nsymbols, each of the plurality of additional sequences consisting of therepeating first codeword of the first cyclic position code, the methodincluding: obtaining, from each of the additional sequences and at aposition corresponding to the at least one point, a respectiveadditional subsequence of length w symbols; mapping each of theadditional subsequences to a respective codeword of the first cyclicposition code; and determining an offset, in the correspondingadditional sequence, of each codeword thus obtained, and therebydetermining the respective additional offset.
 17. The method of claim 16further including deriving a plurality of additional differences betweenpairs of the additional offsets.
 18. The method of claim 17 wherein eachof the plurality of additional sequence is represented by a respectiveone of a plurality of additional patterns disposed or formed on or inthe first substrate, the method including obtaining the correspondingadditional subsequence by detecting or sensing at least part of thecorresponding additional pattern.
 19. The method of claim 18 includingdetecting or sensing w symbols in each of at least w separate sequences.20. A method of determining a first coordinate value of at least onepoint with respect to a plurality of first sequences, each of the firstsequences consisting of a repeating first codeword of a first cyclicposition code, the first cyclic position code having length n₁ andminimum distance, d_(min) ₁ the method including: obtaining, from eachof h₁ the first sequences and at a position corresponding to the atleast one point, a respective first subsequence of length w₁ symbols,where h₁≧2 and w₁≧n₁−d_(min) ₁ ; mapping each of the first subsequencesto a respective codeword of the first cyclic position code; determiningan offset, in the corresponding first sequence, of each codeword thusobtained, and thereby determining a respective one of a plurality offirst offsets of the at least one point; deriving, for each of h₁−1pairs of the first sequences, a difference between the correspondingpair of first offsets, and thereby deriving a respective one of aplurality of first differences; and deriving, from the plurality offirst differences, the first coordinate value.
 21. The method of claim20 further including interpreting at least one of the first differencesas a digit of the first coordinate value.
 22. The method of claim 20further including interpreting at least one of the first differences asa marker separating the first coordinate value from an adjacentcoordinate value.
 23. The method of claim 20 wherein w₁≧n₁−d_(min) ₁ +2.24. The method of claim 20 wherein w₁≧n₁.
 25. The method of claim 20wherein the first cyclic position code is a simplex code.
 26. The methodof claim 20 wherein the first cyclic position code is a code listed inTable 3, or is the reverse, the complement, or the reverse complement ofa code listed in Table
 3. 27. The method of claim 20 further includingdetermining a second coordinate value of the at least one point withrespect to a plurality of second sequences, each of the second sequencesconsisting of a repeating second codeword of a second cyclic positioncode, the second cyclic position code having length n₂ and minimumdistance, d_(min) ₂ the method including: obtaining, from each of h₂ thesecond sequences and at a position corresponding to the at least onepoint, a respective second subsequence of length w₂ symbols, where h₂≧2and w₂≧n₂−d_(min) ₂ +1; mapping each of the second subsequences to arespective codeword of the second cyclic position code; determining anoffset, in the corresponding second sequence, of each codeword thusobtained, and thereby determining a respective one of a plurality ofsecond offsets of the at least one point; and deriving, from theplurality of second offsets, the second coordinate value.
 28. The methodof claim 27 further including interpreting at least one of the secondoffsets as a digit of the second coordinate value.
 29. The method ofclaim 27 further including interpreting at least one of the secondoffsets as a marker separating the second coordinate value from anadjacent coordinate value.
 30. The method of claim 27 further includingderiving, for each of h₂−1 pairs of the second sequences, a differencebetween the corresponding pair of second offsets, and thereby deriving arespective one of a plurality of second differences; and deriving, fromthe plurality of second differences, the second coordinate value. 31.The method of claim 27 wherein w₂≧n₂−d_(min) ₂ +2.
 32. The method ofclaim 27 wherein w₂≧n₂.
 33. The method of claim 27 wherein each of theplurality of first sequences is represented by a respective one of aplurality of first patterns disposed or formed on or in a firstsubstrate, the method including obtaining the corresponding firstsubsequence by detecting or sensing at least part of the correspondingfirst pattern.
 34. The method of claim 33 wherein successive symbols ofeach of the first sequences are represented by successive parts of thecorresponding first pattern arranged in a substantially linear fashion;and the first patterns are arranged in a substantially parallel andspaced apart fashion.
 35. The method of claim 33 wherein each of theplurality of second sequences is represented by a respective one of aplurality of second patterns disposed or formed on or in the firstsubstrate, the method including obtaining the corresponding secondsubsequence by detecting or sensing at least part of the correspondingsecond pattern.
 36. The method of claim 35 wherein successive symbols ofeach of the second sequences are represented by successive parts of thecorresponding second pattern arranged in a substantially linear fashion;the second patterns are arranged in a substantially parallel and spacedapart fashion; and the first patterns and second patterns are arrangedin a substantially orthogonal fashion to each other.
 37. The method ofclaim 27 wherein the first coordinate value and second coordinate valuetogether define a two-dimensional coordinate.
 38. The method of claim 27wherein the second cyclic position code is a simplex code.
 39. Themethod of claim 27 wherein the values of n₁ and n₂ are the same.
 40. Themethod of claim 27 wherein the values of n₁ and n₂ are not the same. 41.The method of claim 27 wherein the first and second cyclic positioncodes are the same.
 42. The method of claim 27 wherein the first andsecond cyclic position codes are not the same.
 43. The method of claim20 wherein the second cyclic position code is a code listed in Table 3,or is the reverse, the complement, or the reverse complement of a codelisted in Table
 3. 44. A method of determining a first offset of atleast one point with respect to a first sequence of at least n symbols,the first sequence consisting of a repeating first codeword of a firstcyclic position code, the first cyclic position code having length n,the method substantially as herein described with reference to thedrawings.